Geometry Help! Confused

Question

A stop sign — a regular octagon — can be formed from a 12-inch square sheet of metal by making four straight cuts that snip off the corners. How long, to the nearest 0.01 inch, are the sides of the resulting polygon?

Answer 1

Sketch your octagon in a square.  Notice that each of the corners to be cut off are 45-45 degree right triangles.

(The external angles in a regular polygon are 360/n.  360/8 = 45.

What you really need is the octagon's side length.

One side of the square includes one side of the octagon (s) and one leg of each of two isosceles right triangles which have (s) as their hypotenuse.  The legs have equal lengths (a).

Use the Pythagorean theorem:

s² = a² + a² = 2a²

s = a√2

a = s/√2

The total length of one side of the square is s + 2a, so:

s + 2a = 12

s + 2s/√2 = 12        multiply top & bottom of the fraction by √2 to clear the radical from its denominator.

s + 2s√2/2 = 12       multiply by 2

s + √2s =  12

s(1+√2) = 12

s = 12 /(1+ √2)

s = 4.97 ≈ 5.0